In the spirit of Soave [1]} and Jeanjean, Le [2]}, we see that the presence of the mass subcritical terms \(\alpha _1|u|^{p-2} u, \alpha _2|v|^{p-2}v\)  causes a convex-concave geometry of \(I|_{S(a_1)\times S(a_2)}\)  if \(\alpha _1,\alpha _2>0\)  and \(a_1,a_2>0\)  are small.
In view of this, if \(\alpha _1,\alpha _2>0\) , it is natural to introduce a suitable local minimization problem:
\(\begin{aligned}&m^+(a_1,a_2):=\inf _{(u,v)\in \mathcal {P}^+_{a_1,a_2}}I(u,v)\\&=\inf _{(u,v)\in \mathcal {P}_{a_1,a_2}}I(u,v)=\inf _{(u,v)\in V(a_1,a_2)}I(u,v)<0,\end{aligned}\) 
